Problem 35 - Introductory and Intermediate Algebra for College Students 4th

Question

Use the addition property of inequality to solve each inequality and graph the solution set on a number line. $$y+\frac{7}{8} \leq \frac{1}{2}$$

Step-by-Step Solution

Answer

The solution to the inequality is \(y \leq -\frac{3}{8}\). This is represented on the number line as a shaded region to the left of -3/8, including -3/8.

Step 1: Isolation of variable y

Start by subtracting \(\frac{7}{8}\) rom both sides of the inequality to isolate y on one side: \(y+\frac{7}{8} - \frac{7}{8} \leq \frac{1}{2} - \frac{7}{8}\)

Step 2: Solving the inequality

Perform the subtraction keeping in mind that operations must be done on fractions: \(y \leq \frac{1}{2} - \frac{7}{8} = -\frac{3}{8}\)

Step 3: Graphing the inequality on a number line

Draw a number line, mark the point -3/8, and shade to the left of -3/8 because y is less than or equal to -3/8. Since the inequality includes 'equal to', also make a dark dot or closed circle at -3/8 on the number line to indicate that -3/8 is included in the solution set.

Key Concepts

Addition Property of Inequality

Understanding the addition property of inequality is crucial in solving inequalities effectively. This property states that you can add or subtract the same number from both sides of an inequality, and the direction of the inequality will not change. For example, if we have an inequality like \( a < b \), after adding \( c \) to both sides, it remains true (given that \( c \) is a real number): \( a + c < b + c \).

In our exercise, we have \( y + \frac{7}{8} \leq \frac{1}{2} \). To begin solving for \( y \), we need to eliminate the \( \frac{7}{8} \) on the side where \( y \) is. By subtracting \( \frac{7}{8} \) from both sides—honoring the addition property of inequality—we maintain the balance of the equation and progress towards isolating \( y \).

Number Line Graphing

Graphing inequalities on a number line helps visually represent the solution set. To graph an inequality on a number line:

• Begin by drawing a horizontal line and evenly spacing out numbers that include the known values from the inequality.
• Place a dot on the number that represents the boundary of the inequality; a closed dot means the number is included in the solution (\( \leq \) or \( \geq \)), whereas an open dot means it is not included (\( < \) or \( > \)).
• Shade the portion of the line that represents the set of numbers satisfying the inequality.

In this case, because the inequality \( y \leq -\frac{3}{8} \) includes the value \( -\frac{3}{8} \), a closed dot is placed there. Then, because \( y \) is less than or equal to this value, the line to the left of the dot is shaded to show all possible values of \( y \) that make the inequality true.

Isolation of Variable

The concept of isolation of a variable in an inequality is akin to doing so in an equation. It means manipulating the inequality in such a way that the variable you are solving for is alone on one side. This often involves using the addition or subtraction properties of inequality, and sometimes involves dividing or multiplying, provided you follow specific rules for those operations with inequalities.

For instance, in our example, we isolated the variable \( y \) by subtracting \( \frac{7}{8} \) from both sides. It is essential to keep the inequality balanced, which means any operation performed on one side must be mirrored on the other. Upon isolating \( y \), we are left with a simplified form \( y \leq -\frac{3}{8} \), which is ready for graphing. It's important to always remember if you multiply or divide both sides of the inequality by a negative number, the direction of the inequality must be reversed to maintain the truth of the statement.